Math 132: Smooth Manifolds, Spring 2017

Summary of Lectures

Here are some notes summarising the material we have covered each week. Let me know if you find any mistakes!

• Week 1
• Week 2
• Week 3
• Week 4
• Week 5
• Week 6
• Week 7
• Week 8

Supplementary Exercises

• Week 1
• Week 2
• Week 4
• Week 5

Test Yourself Quiz

• Week 1 Solution
• Week 2 Solution
• Week 3

Resources

The following is a list of resources (textbooks, online notes, similar courses) that you may find useful and/or interesting. Caution: I recommend finding a resource that you like and sticking with it - jumping back and forth between resources can be a hindrance. This material is standard and each resource says the same thing, though perhaps in a slightly different way.

U = undergraduate level, G = graduate level, R = research level

• Differential Topology, Guillemin & Pollack (U The course textbook.)
• Topology from the Differentiable Viewpoint, Milnor (U/G, excellent though succinct.)
• Introduction to Smooth Manifolds, Lee (G, discusses abstract manifolds and lots of exercises. Available for free with HUID.)
• Introduction to Manifolds, Tu (G, similar to Lee's book. Accessible. Available for free with HUID.)
• Introduction to Differential Forms, Sadun (U/G A nice introduction to differential forms, focusing first on how to use forms in integration.)
• Differential Topology, Hirsch (G Covers a lot of the same material as Guillemin-Pollack but in a more grown-up way. Would be good to read after G-P. Availbale for free with HUID.)
• Differential Manifolds, Kosinski (G/R Focuses on a lot of the mathematics that arose in tackling the gneralised Poincaré Conjecture (in all dimensions). Challenging.)
• Differential Forms in Algebraic Topology, Bott & Tu (G A classic and a must-read for anyone wishing to learn how to compute anything in algebraic topology. Provides an introduction to spectral sequences. Available for free with HUID)
• Dr. Bob's Elementary Differential Geometry, R. Jantzen (U/G Some notes I found online that look like a great introduction to Riemannian geometry for physicists and mathematicians. Quirky but accessible.)
• Morse Theory, Milnor (U/G A must-read for anyone interested in geometry. Morse theory is a simple idea (developed by Marsten Morse (PhD Harvard)) that has repeatedly been used to great effect: in essence, it underlies all significant results in differential topology (and beyond).)